Some Properties of Lie Algebras

[摘要] Traditionally, Lie algebras have been used in physics in the context ofsymmetry groups of dynamical systems, as a powerful tool to study theunderlying conservation laws [1,2]. At present, space-time symmetriesand symmetries related to degrees of freedom are considered. Forinstance, non-trivial Heidelberg algebra arises right in the base ofthe Hamiltonian mechanics. Hamiltonian mechanics describes thestate of a dynamic system with 2n variables (n coordinates and nmomenta), and the other interesting observable physics quantities arefunctions of them. Kuranishi [3] proved that for any finite dimensionalsemisimple Lie algebra L over a field F of characteristic zero there existtwo elements X, Y ∈ L which generate L. Work on simple Lie algebrasof prime characteristic began nearly 75 years ago. Much of this workhas concentrated on the case of restricted Lie algebras (also called Liep-algebras). Robert Zeier and Zoltán Zimborás [4] given a subalgebra hof a compact semisimple Lie algebra g and a finite dimensional, faithfulrepresentation θ of g, then h=g iff dim(com [(θ ⊗ θ)/h])=dim(com[θ⊗ θ]). Bai Ruipu, Gao Yansha and Li Zhengheng [5] proved L be aLie algebra, D be an idempotent derivation. Then the image of D onL, is denoted by I=D(L), is an abelian ideal of L, and the kernel of D, isdenoted by K=KerD is a subalgebra of L . Zhang Chengcheng, ZhangQingcheng [6] Let L be a Lie color algebra. Then adL={adx | x ∈ L} is aLie color subalgebra of End(L), which is said to be the inner derivationalgebra, where a Lie color algebra is a G-graded F-vector space L=⊕g∈GLg with the bilinear product [·,·]: L × L → L satisfying some conditions.Recently, David A. Towers [7] proved there are many interestingresults concerning the question of what certain intrinsic properties ofthe maximal subalgebras of a Lie algebra L imply about the structureof L itself. In this paper, we give some properties on subalgebras andsemisimple of Lie algebras with others concepts.